
Using Grothendieck's identity (e.g. Lemma 3.6.6. in [1]}) we have
\(\mathbb {P} \left( \text{sgn} \left( |X_i^T v | \right) \ne \text{sgn} \left( |X_i^T\beta ^* | \right) \right) = \frac{\arccos \left( \langle v, \beta ^* \rangle \right)}{\pi } \ge \Vert  v- \beta ^* \Vert _2 \ge \eta .\) 
